Quantum Mechanics and the Geometry of Spacetime

SPEAKER 1: Before I introduce our speaker, let me give a little background on this series.
Hans Bethe, one of the giants of 20th century physics, joined the Cornell faculty in 1935 and spent the remainder of his career here, making influential contributions into his 90s. The Bethe Lectures were endowed by the university in 1977 to honor Hans Bethe's service to Cornell, and to bring distinguished physicists here to enrich the intellectual environment of the university.
It's an extraordinary privilege to introduce this year's Bethe lecturer, Professor Juan Maldacena, one of the preeminent figures of 21st century theoretical physics. Professor Maldacena received his early education in Argentina, then obtained his PhD at Princeton in 1996, under the supervision of Curtis Callan, writing a thesis containing a now classic treatment of black holes and string theory.
After a year of postdoctoral work at Rutgers, he joined the Harvard faculty in 1997. In that year, he wrote a single author paper that has arguably done more than any other work to set the course of theoretical physics over the past two decades. Now cited an amazing 10,125 times, the most in the entire INSPIRE database of high energy physics, more even than Weinberg's 1967 "A Model of Leptons."
Professor Maldacena's discovery of AdS/CFT correspondence has been the primary guiding light in string theory and particle theory. But the picture of holography that emerges is so powerful that it now reaches into many other fields as well. Moreover, even setting this paper aside, Professor Maldacena's unrelated work has likewise determined the direction of progress both in string theory and in theoretical [INAUDIBLE].
Professor Maldacena's contributions have been recognized through numerous awards, including the MacArthur Fellowship, the Dirac Medal, and, in 2012, the inaugural $3 million Milner Foundation Fundamental Physics Prize. Since 2001, he has been a professor at the Institute for Advanced Study at Princeton.
Now, before we begin, let me remind graduate students that you'll have an opportunity to meet with Professor Maldacena immediately after the colloquium. And now, please join me in welcoming him.
[APPLAUSE]
JUAN MALDACENA: Well, thank you very much for the kind introduction. It's a great honor for me to be given this Bethe Lectures. So I'll be talking about Quantum Mechanics and the Geometry of Spacetime.
People thought for a while that the geometry of space was Euclidean. Euclid certainly thought so. But then, it was discovered that there were other possible geometries, that we could have hyperbolic space as a possible geometry. And this is a nice picture of hyperbolic space drawn by Escher. Now, if we add the time even to the Euclidean space, we got Minkowski space, and we got the space of special relativity.
And then, putting these two things together, Einstein realized that we could describe the metric of our own spacetime as a non-Euclidean geometry, and that's the Theory of General Relativity, or Geometrodynamics. It's the dynamics of geometry. And this theory makes two very surprising predictions.
The first is the existence of black holes, and the second is the existence of an expanding universe, that the universe expands. And these predictions were so surprising that Einstein himself didn't believe them. I love this phrase that Einstein told Lemaitre, apparently, "Your math is great, but your physics is dismal." Now, I like it, particularly, because this is what physicists usually tell us string theorists.
[LAUGHTER]
JUAN MALDACENA: Now, in both cases, we have this drastic stretching of space and time. And for that reason, we get these surprising predictions. Now, general relativity is a classical theory. It's a classic field theory, similar to classical electromagnetism, in the sense that knowing the spatial geometry at one time, and its [INAUDIBLE], we can determine it in the future. And we know that nature is quantum mechanical, and then, when we have a classical system, the correct quantum description involves a certain procedure called quantization.
However, it is quite difficult to change the shape of spacetime. And what this means is that, for most situations, quantum mechanical effects can be completely ignored. Quantum mechanical effects of the geometry can be completely ignored for most situations. And for an even bigger set of situations, you can actually quantize them, assuming that you have a background geometry, which is fixed. And then, you have small fluctuations of the geometry. And this gives a well defined procedure for quantizing the geometry. And reasonably good in this approximation.
Now, that approximation completely fades at the beginning of the Big Bang. At the very beginning, even before inflation, we really need full theory of quantum gravity to really understand the beginning of the universe. But for now on, we'll discuss this perturbative quantization, so small deviations around the fixed geometry. And we'll see that even that has very interesting consequences.
So that simple treatment of small fluctuations around the fixed geometry leads to also two very surprising predictions related to the two previous ones. One is that those black holes, which were just simple vacuum solutions of the Einstein theory, actually have a temperature. And the second one is that, if you have an expanding universe that this expanding with constant acceleration, then it also has a temperature. In both cases, the formula for the temperature is rather similar.
So in the case of black holes, the temperature is inversely proportional to the size. What this is saying is that the typical thermal wavelength of the metered radiation is of the order of the size of a black hole. This, in particular, implies that black holes, for example, can be white. So you have a black hole that has the size of the wavelength of light, it would look white.
So for cosmology, we have our formula, which is very similar, especially if we express it in terms of the radius of the horizon. So a universe which is expanding with constant acceleration rate has a cosmological horizon. So there is a position such that if something lies beyond that position, you can never see it. And the temperature has a similar form. And as we will see, this last particular temperature is actually very relevant for us.
Now, to explain why it's relevant, I'd like to review the theory of inflation, which says that there is a period of expansion with almost constant acceleration in the beginning of the universe, which is well approximated by this constant expansion. This, according to the classical theory, so classical equations, produces a very large homogeneous universe. So if it was initially homogeneous, these periods of expansion makes it perfectly homogeneous. And if the classical theory was all there was, we would have a completely homogeneous universe. We would have no galaxies, no planets, nothing like this.
However, the fact that we had those quantum fluctuations we talked about imply that there were some temperature and that led to small inhomogeneities in the beginning of the universe, which then seeded the structure we currently see in the universe. So we see this picture, for example, of the cosmic background radiation that has imprinted this initial inhomogeneities.
So we see that quantum mechanics is actually crucial for understanding the large scale geometry of the universe. So we normally think that quantum mechanics is important for small things, right? For atoms, and things like this. But this remarkable, incredibly interesting aspect of nature that actually that the universe at its biggest scales is also determined by quantum mechanics. It's one of these amazing things of physics.
You know, who would have thought that quantum mechanics has something to do with the large scale structure of the universe. It's amazing. And actually the derivation of the spectrum of fluctuations is relatively simple. It's really time dependent harmonic oscillator. I highly recommend to all of you to read this derivation, because it's, I think, one of the amazing things in physics.
Now, what I'd like to emphasize is the fact that these primordial fluctuations are nearly scale invariant. And if the universe had expanded exactly with constant acceleration there would be exactly scale invariant, but the expansion rate was slowing down a little bit and inflation eventually ended, and so they are nearly scale invariant.
What this is saying is that the probability amplitude for having a sudden fluctuation is independent of the size of the fluctuation. So the probability of, let's say, having a fluctuation, which is the size of the universe, and one which is a quarter of the size of the universe, is the same, if both have the same amplitude.
So we can think of these probabilities as determined by the initial wave function of the universe. So the square of this wave function gives us the probability that we see bumps in the CMB with various shapes, various amplitudes. And that is a scale invariant.
Now, scale invariance is something that has appeared in physics a lot. If we decide that if we rescale all distances, then we see the same physics. So a system is scale invariant if looking at the same system at different length scales, we see essentially the same thing. So an example is the electric force or classical electromagnetism.
Now, most of everyday physics is not scale invariant. So for that reason, we have a rather poor intuition for scale invariant systems. Especially for interacting scale invariant systems, we have a poor intuition. But there are several systems that have been encountered in physics that have this feature of being scale invariant.
In particular, condensed matter systems at 2nd order phase transitions or quantum critical points are displayed as a type of scale invariance. In fact, Wilson, here at Cornell, was the one who developed a lot of the theory of such systems. Also the theory of quantum chromodynamics at high energies also becomes nearly scale invariant. So there are several systems that have these features.
Now, systems that have scale invariance sometimes also have an additional symmetrical conformal invariance, which are also some kind of transformations which distort the shape of space while preserving the angles. So this is a picture that Escher drew by making a kind of conformal transformation of an ordinary image. Now, conformal invariance will not be very important for us, so this is just a parenthetical comment.
So let's go back to the scale invariance description. Now, so the universe, in the beginning, if we were treat it in the simplest approximation, is producing for us a scale invariant function of geometry. So the wave function of the universe is a function of the geometry, which is scale invariant. Now, presumably, if we did it more exactly, we'll get some other wave function which is computed better and better. But we can wonder whether this feature of scale invariance is perhaps exact.
Now, in statistical field theories, we can get scale invariant functions of geometry, and that corresponds to taking a statistical field theory and putting it on a general geometry. So we can put a system of spins, for example, at the critical point on a general geometry, and that will give us the partition function of that, or the thermal free energy, if you wish, of that field theory at that critical point will you give us a function of this geometry. So this is a partition function of the geometry.
And we can wonder whether the wave function of the universe could be a function of this kind, a function of geometry given by such a formula. So the advantage would be that we would have an exact formula for the wave function of the universe.
Now, we don't know whether this is true. Some are of the opinion that this might be the case, and some are of the opinion that this couldn't possibly be the case. In fact, not only that, but we actually have a specific example that Tom Hartman, who is here, has proposed, where there is an example of a relationship of this kind where the theory of gravity is, unfortunately, not Einstein gravity, but the slightly somewhat strange theory of higher spin gravity. So even with this example, we don't yet know whether it's a general feature or not.
Now, the interesting thing is that this seems to, and did, work, and it works, for hyperbolic space. So this previous formula that we discussed, which we are not sure whether it's true for an expanding universe, we think it's sure for some other kinds of universes. OK? So let me discuss those other kinds of universes.
So what I said so far is the motivation for why you might be interested in finding such relations, these type of formulas. And now, I'll be discussing some more specific situations where we know that these kinds of formulas are correct.
So this is the expanding universe. So we have the spatial coordinates, and we have the time coordinate. And as time evolves, as time goes forward, the size of the spatial coordinates, the proper size, is increasing. Now, if we made this direction spacelike, so we changed just this sign, I am just calling the coordinate rho now, this space described by this metric is called hyperbolic space. So the expanding universe with constant acceleration is very closely related to hyperbolic space. I'm not saying it's identical. These two have different signs for the curvature, but they are closely related.
If we now take one of these coordinates-- so these were all the spacial coordinates-- when we take one of the spatial coordinates and we make it timelike, then we get a space which also has a time signature. It has a one-time direction, so it could be a space in which we can consider a physical theory. And this is called Anti De-Sitter. So this one is called De-Sitter. This is called Anti De-Sitter. This one has positive curvature. This has negative curvature. This is the simplest solution with positive curvature of Einstein's equations. This is the simplest solution with constant negative curvature. And as you can see, they are very closely related to the simplest non-Euclidean geometry, which is the geometry of hyperbolic space.
So the statement is that the wavefunction of the universe in Anti De-Sitter space now, so in hyperbolic space, can be computed in terms of a quantum field theory that lives on the boundary of this space. So that's the idea.
So this is called a gauge/gravity duality, or gauge/string duality, or AdS/CFT, or holography. So it has various names. All these names, essentially, they know the same thing, which is the following. So these dualities are a kind of bridge that connects two kinds of theories.
So one is a quantum field theory. So a theory of quantum interacting particles with quantum many-body systems, with theories where we have a dynamical space-time. So theories like general relativity or string theory. And the idea is that the string theory or the gravity theory lives in the interior of some geometry, and the fields will live on the boundary. And we'll emphasize that later.
So the original argument for this was some arguments from string theory. I will not give you all the background to these arguments, I will just give you a little sketch of the argument.
So the idea is that there is this theory called a string theory, and it lives in 10 dimensions, and it contains gravity, it contains strings, and it also contains some objects which are called 3-branes. These are some membranes that exist extended along three spatial dimensions plus the time direction.
And Polchinski gave a very precise string theory description of these branes, in terms of open strings and so on. And previously, Horowitz and Strominger had given some classical solutions, classical gravity solutions, which describe how the tension of these branes-- so the somehow heaviness of these branes-- curves the geometry around them, and they produce an object which is possibly like a black hole, except that it extended along three of the spatial dimensions. So those are called black 3-branes.
And the argument was that, if you took the system and you take the lower energy limit of the systems, then this description that Polchinski found as reduced at the lower energies to a certain quantum field theory, which is a version of quantum chromodynamics. So it's a version of quantum chromodynamics, which had certain special symmetry called supersymmetry, and that was the same as the-- and the lower energy limit in this side reduces to two parts, one is very long distances away from the branes-- that's not too interesting-- and the other one is very close to the horizon of this black brane. We have a geometry, which is Anti De-Sitter 5 times the 5 sphere.
So this is the Anti De-Sitter space we were talking about before, except their five-dimensional. And the idea is that these two theories were the same thing. So there was an initial example that was derived from the original strength theory, but then people realized that, well, we can have more general cases of this relationship, and that this broadly general feature of quantum gravity on hyperbolic space, as long as we are dealing with a consistent theory of quantum gravity, whether it comes from string theory, or from some other theory.
So in order to understand this relationship, it's convenient to take a step back. And I'll discuss a couple of features of Anti De-Sitter space, and also a couple of features about dynamics in scale invariant theories, which will make this relationship seem a little more plausible.
Now, this is the metric, again, of Anti De-Sitter space. And I want to draw your attention to this function that sits in front of the time direction. The function that sits in front of the time direction gives us, well, the gravitational redshift. And we can think of this as a general relativistic analog of the gravitational potential. It's just a Newtonian potential. So this is the general relativity version of the Newtonian potential.
And we can make a plot of this. So it's a function of rho, so the horizontal direction here is rho. We have this potential, which is very small here and very large on the right. So if you put the particle in this space, and you put it at rest at some value of the rho coordinate, then, well, it's like putting a particle in a gravitational potential that has this shape, this particle will move to the left. So particles in this space are really pushed to the left. So it's pushed towards the smaller-- back to the places where the gravitational potential is smaller.
So in some sense, if you were standing on this space at some fixed rho value, you would feel as if you were in a constant gravitational field. Of course, if you're not standing on the floor, you will just fall, and you would feel you are just simply executing free fall.
So that's what happens in Anti De-Sitter space. Now, let's forget that for a second, and talk about dynamics in scale invariant theories.
So imagine that you have a scale invariant theory, and you make a small deformation of the geometry, some external deformation of the geometry. So that will create some particles, some vibration. So you can think, I mean, you have a system and you bang it with a hammer, you create sound waves, right? So that's an example of what I'm talking about.
But we can have any theory that can have any kinds of interactions. We can always make a small perturbation, and we'll create some object, OK? And the question is, how do we describe that object?
So in the case of sound waves, we know that the quantum mechanical description is that we create a phonon and that behaves like a particle, and so on. It's a massless particle. But now, what is the general description for an arbitrary theory? How do we characterize it?
Well, in theories that have a mass gap, so where we have massive particles, then we know will create an excitation, and that excitation will be described by three coordinates, three positions. OK? Of course, we should also specify the mass of the particle.
Now, in these scale invariant theories, in addition to specifying the position, we should also specify the size of the particle. So there's one new variable that we should specify. Then, there is also another quantity which is similar to the mass, which is called the scaling dimension of the object. I won't discuss it in detail. But the crucial point I want to emphasize is the fact that we need one more coordinate. So specifying the center of mass position is not enough. We also need to specify the size. OK?
And sometimes people call these type of excitations the excitations of the scale invariant theories and particles. And if you ever read the first pages of Landau and Lifshitz, the "Mechanics" course, the first book in the series, there they derive the action for a free massive particle just from the symmetries of the problem.
Now, we can do exactly the same thing for the scale invariant theory. So we have the coordinates, let's say, the three positions and the time coordinate, parametrized in terms of an arbitrary parameter tau. And then, we also have the size of the system. So we have these coordinates. And then, we write down the simplest action that has all the symmetries. So the symmetries and their translations, symmetries under rescaling of both X and C. And the simplest action has this form OK?
So the ordinary symmetries and the translations, and so on, imply they should be a function of X dot squared. And then, the symmetries and the rescaling imply the rest of the form of the action.
OK, and it turns out that this action is exactly the same as the action for a particle in Anti De-Sitter space, where the coordinate rho that we discussed before is related to C in this way. So C equal to 0 corresponds to rho equal to plus infinity. So recall the variable rho we defined before here in this formula. OK? So C equal to 0 corresponds to this region, and C equal to infinity to that region over there. OK.
So we conclude that this unparticle in 3+1 dimensions is the same as an ordinary particle in 5 dimensions.
Now, this is the same that I was saying before, but in terms of pictures. So we have the boundary theory. So these are the spacial directions along the boundary that have been drawn here. And we have two objects on the boundary. So one object is simple elementary excitation, which has some size. This is the red object. And the blue one is exactly the same, but that has been rescaled, has been made a bit bigger. And it is that this red object corresponds to a particle that is at some radial position in the interior geometry. And the blue one is exactly the same particle, but just moved deeper into the interior. It's further down in the gravitational potential.
OK, this discussion makes it look like any CFT, then, gives rise to some theory in the interior. So we said that any excitation can be viewed as a particle in Anti De-Sitter space. That's completely general. I mean, this argument I just made with pictures can be made with math. I mean, into a complete equivalence of representations and group theory, and so on. So there is no hand waving there. It's completely correct.
But now the question is whether it is true that any conformal field theory gives rise to some kind of quantum gravity theory. And the answer is both yes and no.
So it is yes in the sense that, well, yes, but it could be a very strongly coupled gravity theory, which we can't really define in an alternative way. And so we don't know really what we are talking about. So it's yes in some sense.
But if we want to say that, by a gravity theory we mean something more specific, like a theory that, first of all, is weakly coupled so that the graviton interacts weakly with itself, and second, that the equations that describe internal waves are the equations of Einstein gravity-- so the ordinary Einstein action, the secondary relative actions or the action which is the usual Einstein equations-- then we need more conditions. So what I will now describe is what extra conditions we need in order to have these two properties.
So first, we'll ask the first condition. So first, I need to say that, if you have a field theory, you always have some operator that measures the local energy, or the stress tensor operator. That's an operator in the conformal field theory that's one of the ones you will have if you have any theory. Any quantum field theory, you will have this stress tensor operator. And this stress tensor operator gives rise to massless spin two particle in Anti De-Sitter space.
So the elementary action of the stress tensor on the boundary creates in the interior, through the mechanism we've seen before, a spin two particle which happens also to be massless. So the fact that it has spin two is related to the fact that this stress tensor also spin two, has these two indices. And the fact that it's massless is related to the scaling dimension of the stress tensor. So the fact that the stress tensor measures energy as units of energy divided by volume. Just the simple facts give rise to-- through the translation, that group theory statement that I mentioned before, that operators on the boundary create states that can be viewed as states in-- gives rise to a massless spin two particle.
Now, in order to really have a particle, something that we associate to a particle, we need that the interactions are relatively weak. So anything that we want to recognize as a particle is something that is relatively weakly interacting. And weak coupling means that it is difficult to change the metric. Now, the stress tensor is the same-- I mean, the two point function of the stress tensor is the same, based by definition, the derivative of the partition function with respect to the metric, so with the response of the system to a change in the geometry where the system is sitting.
So that's in the quantum field theory. And if you have a quantum field theory with a certain number of fields, that's roughly proportional to the number of fields you have in your theory. If the theory is very strong interacting, it's a way of defining an effective number of fields that you have in your theory.
Now, this same calculation can be done in the bulk theory, using the Einstein action. And I said that there was this relationship between this and the wave function of the universe, so we can calculate the wave function used in the Einstein theory. We can calculate the small changes of this wave function in this way, and we get that the result depends on the Newton constant. So it depends on the normalization of the action in Einstein's theory, in units of the radius of Anti De-Sitter.
So Anti De-Sitter space has a certain radius of curvature, and this is what we get. So when the theory is weakly coupled, this quantity is very, very large. So this is the Planck length. This is the radius of Anti De-Sitter space in Planck units. And for the gravity theory to be weakly coupled, we need these to be very large. And if that's very large, it means that the number of fields we need in the boundary theory is a very large.
So whenever we are talking about weakly coupled theory in the interior, we are talking about the theory on the boundary that has a large number of fields.
Just to give some numbers. So if we have the very simple scale invariant theory that Wilson here was studying, that number would be 1. So that theory does not have a weakly coupled gravity theory. It might have a strongly coupled gravity dual, which we don't know exactly what it-- well, we do know a little bit about it. I mean, it's one of these higher spin gravity duals, but we don't know it very well.
The theory of QCD has this number roughly of about 8. If we look at the early universe, this number would be 10 to the 12. So here, the number is now computed using gravity. In the present universe, which is also entering a period of constant acceleration, the number would be 10 to the 120.
So depending on the application that we use, we'll use these formulas that will have these numbers which range vary widely. OK?
So the conclusion is that, if the bulk gravity theory is weakly coupled, then the boundary theory because a large number of fields.
Now, saying that it's weakly coupled means that the interaction between the gravitons is going to be small. But it will not necessarily be that of the Einstein theory. And the rest of the interactions are also not necessarily those of the Einstein theory. In order to get that, we need an extra condition.
So all these theories that are different from the Einstein theory, but still weakly coupled, they also have higher spin light particles. And so we really need to give a large mass to all these higher spin particles in the bulk. And for that, we need very strong interactions of the theory on the boundary. So theories on the boundary which are weakly coupled, like the theories we are normally used to, which are weakly coupled-- chromodynamics, for example, at very high energies, which is weakly coupled, or even the Wilson-Fisher Fixed Point, and even the Russian version of the Wilson-Fisher Fixed Point, which has a large number of fields-- they all are theories which are weakly coupled on the boundary, and that implies that the theory on the bulk contains these extra light higher spin particles that we don't want. And the only way to get rid of those higher spin particles is to make the boundary theory strongly interacting.
So for very strongly interacting theories, we expect that all these higher spin particles are very heavy. However, the stress tensor continues to give rise to a massless spin two particle, which is the one of Einstein's theory. And in this case, we expect to get the theory of Einstein gravity, plus perhaps extra particles with spin less than two, like matter fields that, say, spins 1/2 fields, spin 0 fields, and so on, spin one fields. OK.
And the process of exactly how this emerges in general is not completely understood, but we know some examples where it does. So I'm going to briefly say what happens in one particular example.
So the example I mentioned before of supersymmetric quantum chromodynamics. So supersymmetry is a special symmetry that makes theories easier to calculate. That's why this example has been studied more than other examples.
In this example, we can look at the spin for particle spin 4 state as we bury the coupling. So we have the cupping of the Yang Mills theory, and the effective coupling is-- well, Yang Mills couplings times 10, so lambda is the effective coupling. If the company is very weak, we can calculate the mass using the weakly interacting theory of gluons, and we find the mass with a zero coupling. And then, it starts increasing a little bit when we increase the coupling.
On the other hand, at very strong coupling, we have-- the gauge/gravity duality tells us that we have an Anti De-Sitter space whose radius of curvature in string units is related to the coupling through this formula. I mean, I haven't explained this formula. This formula can be derived using that argument based on branes string theory.
But what this formula is saying is that, as we make the coupling very large in the boundary theory, when the gluons are very strongly interacting, then the radius of the space in string units is very, very large. So the string units are measuring the spacing units of the size of the string. So the size of the massive string mods.
So in string theory, we do have these higher spin states, but they have a mass of [INAUDIBLE] imprecise of the string size. That's what really this means. This is the mass of the higher spin particles, 1 over ls is the mass of the higher spin particles. And this is very large in a strong coupling. So that's where this lambda to the 1/4 part comes here. So we can calculate this easily at very weak or very strong coupling. So this we can calculate using one of the sides of the duality, and this the other side. So we calculate using the gravity, and this we calculate using the Yang Mills side. OK.
And in this particular case, we can-- through an incredible tour de force, the complete formula here was calculated, using symmetry of integrability. And actually, this uses some techniques that go back to Hans Bethe. I should have put Hans Bethe here in the beginning. When he saw the Heisenberg spin chain, he realized that the spin chain had extra-- well, this was actually realized later-- that it had extra symmetry properties.
And this led to a series of techniques that were used mainly in condensed matter physics, and sometimes into the measurement field theories. But using those techniques, one can actually do this computation of the exact scaling exponents in this scale invariant theory in four dimensions.
So what these people have done is, using Bethe's technique, they computed the Wilson scaling coefficient, that Wilson could only calculate approximately. So in any particular theory, there is a special four-dimensional theory where we can calculate the Wilson scaling coefficients, which are, as we said, related to the masses of particles in the bulk, and they, indeed, have the form expected from the gauge/gravity duality.
The reason that this is related to spin chains is that in the larger limit, the limits of the large number of colors, the gluons are interacting only with gluons that have colors which are correlated to their own. So a gluon has a color and an anticolor, and then there is a second gluon that has its own color and anticolor, and only when the color of these, and the anticolor of these, are the same, they interact strongly. Otherwise, they don't.
So they form little chains where the colors are correlated in this way. And these little chains gives rise to become a strong coupling, the strength to live in Anti De-Sitter space. And actually, in fact, this relationship holds at all values of the coupling. And that's what these people have shown, that this picture is really connected. And we really can go continuously between these change of gluons and these fundamental strings.
Now, I should mention, as a parenthetical comment, that strings have already been observed in nature. Sometimes people say that, well, string theory, you know, you do these things, it's never been observed in nature. But strings have actually been observed in nature. They are the strings of QCD, so we explain them as the strings of QCD. Not only observed in nature, they are used to explain the QCD observables. And we don't yet know how to go between the fundamental gluons of QCD and the strings of QCD. I mean, it can be done using numerical simulations, but not arithmetically. But this is an example of a QCD-like theory where you can actually continuously go between these two pictures.
So we also expect that the picture similar to this would be true in quantum chromodynamics, that there might be, at least in the larger limit of quantum chromodynamics where the number of colors is not three, but is much larger than three, we could have a certain five-dimensional string theory. That string theory hasn't been found completely yet.
OK, so there are some practical applications of this relationship. And the practical applications consist-- they basically use the following fact, that strongly coupled field theory problems translate into simple gravity problems. So we said that, in order for the gravity description to seem to be the Einstein one, we needed the boundary theory to be very strongly coupled. And so, if you have a situation where you have a strongly coupled field theory with a gravity dual, then you can do a simple gravity calculation that lets you solve this strongly coupled field theory.
And this has been applied to various problems. Well, first as models for QCD, by replacing QCD by the supersymmetric version of QCD. So it's a kind of approximation where you change your theory to deal with a toy model that is better understood. Or also for getting some intuition for what strongly interacting condensed matter systems could be doing. Of course, in condensed matter systems, or even in QCD, we don't have n very large. We don't have couplings extremely large, by at least it could give us some intuition of what is going on.
Now, I'd like to answer the first question of why could strong coupling simplify the problem, and whether we have other situations in physics where strong coupling simplifies the problem a bit. And actually, there is a very simple case where some amount of strong coupling simplifies the problem.
So for example, if you have a gas of particles that is somewhat interacting, then at long distances, the dynamics is described by hydrodynamics. And on the other hand, if you had exactly zero interactions, then this fluid would have infinite viscosity. So some interactions are important for the fluid having been described by hydrodynamics.
So we don't know how this exactly works in gravity. So we know these examples. So gravity is analogous to this hydrodynamic approximation, where we keep some low energy modes-- low energy, the use of freedom. Except that it's not necessarily statistical, but also it makes sense quantum mechanically. And in some cases, actually, as we'll discuss later, this analogy's actually precise.
Now, an interesting set of problems appears when you consider thermal configurations in Anti De-Sitter space. So if you raise the temperature, then you can have a black hole in the interior, which has certain temperature. And that will be the thermal equilibrium situation in these gravitational theories. So you will have these situations.
And in the boundary theory, these are described by gas of particles at finite temperature in the boundary. So you have a fluid on the boundary, kind of quark-gluon plasma, which, at finite temperature, it has some entropy. And in particular, the entropy of the black hole, and gravity is computed by the area of the horizon, is given by the log of the number of states of this plasma. So this gives us statistical interpretation to the entropy of black holes.
So the entropy of black holes is somewhat mysterious, because the entropy used in thermodynamics-- I mean, using the formula for the temperature we mentioned in the beginning, we can derive this formula for the entropy. But the black hole is just a single solution of the gravity equations. So what is this entropy counting? What is the microscopic origin of the entropy?
So in all systems in physics, when you measure some entropy in the lab, you would like to assign to it some statistical interpretation, in terms of the things that are moving and are giving rise to this temperature. So in this case, we have a description of this entropy in terms of the fields that live on the boundary. So in terms of those degrees of freedom, we have a clear description of the entropy.
Also when you think about gravity in the bulk, it's not clear whether the evolution of this black hole is really going to respect unitarity, whether it will be unitary. Because it seems that things can fall into a black hole and that information cannot come back out. On the other hand, when you think about it from the boundary point of view, you have no loss of information. You have this fluid that, if, in principle you could describe exactly, with no loss of information.
So if you accept this gauge/gravity duality, then there is no information loss due to black holes when we see the black hole from outside. On the other hand, the description of the interior is not yet well understood, and there is some debate on what the gauge/gravity duality implies for the interior of black holes, with some authors saying that there are drastic deviations from GR, and some saying that perhaps there aren't. But the truth is that no one has given a very precise description of their interior of the black hole from the boundary point of view.
On the other hand, the black hole seen from the exterior gives rise to a bunch of interesting properties. So we said that, if we have the field theory at finite temperature, we have a thermal system. And in the bulk, in the interior, this is described by a black brane. So that's like a black hole which is extended in space, whose horizon is extended in space. And then, small ripples, long distance ripples on this black brane, correspond to the hydrodynamic modes of this plasma, of the plasma on the boundary. And the absorption of the ripple into the black hole-- so the fact that it falls into the black hole-- is related to the fact that, on the boundary, we have dissipation and viscosity.
In fact, we can calculate the dissipation and viscosity of this plasma by solving the wave equation in this background. So in general, in a strongly coupled theory, it's very difficult to calculate such coefficients, the viscosity and so one of the fluid. On the other hand, here it can be calculated simply by solving the wave equation. This is an example where a complicated problem translates into a simple problem.
And finally, it can be shown that the Einstein equations in the near horizon region actually reduce to the hydrodynamic equations on the boundary theory. So the long distance part of the Einstein equation, near the black hole horizon, reduces to the hydrodynamic equations. In general, relativistic hydrodynamics, but you can take the nonrelativistic limit and you get the standard Navier-Stokes equations.
Well, in fact, when people work this out-- so this is a particular example where this duality was useful, because when they have this for some special cases-- when these authors have this for some particular cases-- they found they didn't match. These two equations didn't match with theories with anomalies, with something called anomalies, they actually didn't match. And then, it turned out that the hydrodynamics needed some corrections when you had anomalies. And those corrections were present even in theories that didn't have gravity duals. This an example where you learn something from the duality that, then, you could apply to other systems.
OK, so now I'd like to focus on one particular feature of this duality. And this is a connection between the entanglement and geometry. And so the pattern of entanglement in the quantum state and the boundary theory can translate into geometrical features of the interior.
So for example, you can take the field theory on the boundary, and you can consider separation of the boundary theory. And in a quantum field theory, the degrees of freedom of this region are entangled with the use of freedom outside. And this is related to the fact that black holes have a temperature, and so on. So it's an important fact of quantum field theory.
And using the duality, now we have an extra dimension, this direction that goes down orthogonal to the blue line, and we can calculate the entanglement entropy-- so the amount of entanglement we have-- by looking at the minimal area surface in the bulk. And there is a formula which is very similar to the formula for black hole entropy, the Hawking-Bekenstein formula that describes this entanglement. And this has been useful for studying various aspects of this for understanding how the use of freedom are entangled in quantum field theory.
Now, this is particularly interesting for a solution, which is the simplest black hole solution in Anti De-Sitter space. And so far, I've been describing the black hole has seen from the outside. But when you extend the solution towards the interior as much as you can, you find a solution which is so-called eternal black hole. So this is the maximally extended black hole solution in Anti De-Sitter space.
So it contains two exterior regions. This region here looks like a black hole in Anti De-Sitter space, as seen from the outside. And this one here also looks the same. But this is a second Anti De-Sitter space, not related to the first one.
So this can be interpreted as an entangled state in two not interacting CFDs. So in other words, we think of this geometry, which is actually connected here through the interior, as arising from two of these connected field theories. So two theories whose Hamiltonians are completely disconnected, but which are in a particular entangled state, which is a state closely related to the thermal state. In fact, if we forget about one of the sides, then the state of this theory looks like a thermal density matrix. But the state of bulk theories is actually an entangled state.
So we see that we have this geometric connection coming from entanglement. So this blue line is sometimes called the Einstein-Rosen bridge. It exists also in the Schwarzschild black hole solution, which has this other spacetime diagram, I guess. Here, each side asymptotically looks like flat space, empty flat space. These 45 degree lines are the positions of the black hole horizon. They describe the trajectory of light. Light rays here move about 45 degrees in this diagram. So this is the exterior. This is the interior. And this is a second exterior.
We can think of these two exteriors as living in the same spacetime. We can have two black holes that are very separated, but are connected through the interior in this way. And this is a kind of weird solution, which looks like a wormhole. So it looks like there is a very short distance from the exterior of this black hole to the exterior of the other block hole, a very short spatial distance, but you cannot use it to send signals. So if you tried to send a signal from here to this side, the signal will end up at the singularity. So it is like a wormhole, but a wormhole which, if you cross it, if you attempt to cross it, the universe sort of collapses on you and you can't really make it to the other side.
So this solution is strange. It's allowed by the equations. And one might namely be worried that it leads to causality violation, because it's joining a violation of spacial relativity, right? Because it's joining two points that are very far away through these very short geometric connections.
But the fact is that it doesn't And it doesn't, as long as the fields that live on this geometry obey the standard positivity condition. So you cannot use these wormholes for time travel. So they are not the science fiction wormholes, OK?
So the interpretation we are proposing of these wormholes is that they correspond to an EPR pair, so an entangled pair of two black holes, in a particular entangled state. So for this particular entangled state, we have this connection between ER and EPR. I'm using that these-- these people worked on the exactly the same problems in exactly the same year. So both papers were written in 1935-- the Einstein-Rosen and Einstein-Podulsky-Rosen papers. The Einstein-Podulsky-Rosen paper was the one who introduced entanglement in quantum mechanics, and pointed out the existence of entangled states in quantum mechanics.
So what we learn is that large amounts of entanglement can give rise to this geometric connection. And if one accepts the idea of very quantum geometries, then even the spin 1/2 entangled states could be connected by tiny quantum wormholes in some sense. Now, here, the fine print says that this is not very meaningful, because we don't have an alternative way of defining quantum gravity for these cases. This is an extreme case, where the numbers of degrees of freedom is one. It's one of those cases where we cannot say that there is some Einstein gravity solution. So this last sentence is pure speculation.
Now, the conclusions is that I want to emphasize that cosmic scale invariance. So the scale invariance we see in cosmic fluctuations is connected or related to the field theory scale invariance-- to the scale invariance we see in field theory. So the same type of scale invariance. And in fact, it might, in some cases, or in cases with negative cosmological constant, it's indeed the same thing. And the quantum version of hyperbolic space, so the spacetime with negative curvature, is the same as a scale invariant field here on the boundary.
And this has led to toy models for strongly interactive systems that are useful for testing various hypothesis that one can have how about strongly coupled phantom systems. And it gives a complete description of black holes seen from the outside. And we also mentioned the fact that these large amounts of entanglement can give rise to a geometric connection.
Now, for the challenges for the future. So it looks as if we are one "sign" away from understanding the Big Bang singularity. And in fact, there is one interesting example that I mentioned to Tom Hartman and collaborators of perhaps the similar correspondence in the case of the De-Sitter space, but there are no yet examples with Einstein gravity.
We would like to understand classified conformal field theories that give rise to large universes like ours, and would also like to understand better how we get this local theory in the bulk. In particular, the interior of black holes is still mysterious, and we don't understand it, and we should understand it.
And one of the general lessons is that spacetime is an emergent concept, and that entanglement plays an important role in making it work, and that entanglement can generate a geometric connection of spacetime. Thank you.
[APPLAUSE]
SPEAKER 1: Thanks very much. We have time for some questions.
SPEAKER 2: You mentioned that [INAUDIBLE] in AdS/CFT doesn't do so well yet in understanding the interiors of black holes. I was wondering, is that, do you think, because that people haven't figured out how to do the calculations yet, or [INAUDIBLE]?
JUAN MALDACENA: I think that we are perhaps lacking the conceptual idea that tells us what approximation we need to do in the field theory to understand the interior. So it looks like it's likely that we need to make some approximation, then we'll construct the interior.
That's my point of view. Other people who work in this field will tell you a different thing. So if you ask the same question perhaps to Joe Polchinski he would tell you that we need to find a description of the bulk that is as complete as the description on the boundary, and that we'll need extra use of freedom to describe the interior, which are not present in the boundary theory.
SPEAKER 3: [INAUDIBLE]
Well, our problem is that, if we wanted to-- so AdS/CFT can tell you anything that you measure from outside, and can tell you if you-- I mean, it establishes a dictionary between the boundary theory and the bulk, but the bulk quantities are always measured from infinity. OK?
So you can send in something to the black hole, and so on, and we can tell someone who is doing numerical calculations which calculation you need to do in order to compare to gravity for such observables. But if we ask a question about, let's say, an observer who falls in the interior and gets very near the singularity, what numerical calculation do you have to do to calculate the actual curvature near the singularity, well, that we cannot tell the person doing the numerical simulation. So I think it's something more that is needed beyond the numerical simulation.
I mean, at this point, it might be that someone starts doing the numerical simulation and finds some pattern that allows you to figure out what the dictionary should be. So that could well happen. So in the same way that Wilson got the pattern for normalization by doing numerical simulations of field theory, it might be that you start doing simulations of these systems and you find some pattern, and you realize what the central concept should be.
SPEAKER 4: [INAUDIBLE]
SPEAKER 1: Could you repeat the question? [INAUDIBLE]
JUAN MALDACENA: Yeah, the question was related to one of the slides that was whether-- a version of AdS/CFD for De-Sitter space, whether it's true or not true, OK? So this equality in the case of an expanding universe, right? Well, what happened here? Whether this is true or not, all right?
OK, so the arguments that this should be the case is all the calculations you do in perturbation theory will have this property, are consistent with this idea. But that consistency can also be explained by the mathematical fact that those calculations can be obtained by an analytic continuation from the Anti De-Sitter case. So the two are not too different.
Now, you can view this as evidence for the reality, or as evidence against, that you are not testing anything, it's just a consequence of the analytic continuation. So people who say yes view this as evidence for, and people say no say, well, there's no evidence.
Now, the evidence against this is the idea that, in general, we expect Anti De-Sitter space to be unstable-- sorry, De-Sitter space to be unstable. That any expanding universe, like the one we're living in, will eventually decay into some other vacuum. Perhaps one with zero cosmological constant. That solutions with [INAUDIBLE] cosmological constant are inherently unstable.
And so, then, when you go into the far future, you don't have a simple De-Sitter space. You have something very complicated that contains all these decayed products. So that's the main argument for the no here.
There are counter-arguments to this. Well, I don't know if I should go into the counter-arguments, but the counter-argument is that this could still be true, but it's computing the improbable state where the universe has not decayed. So it gives you an exact formula for that improbable state.
SPEAKER 5: But, then, how do you advance this idea further?
JUAN MALDACENA: Well, I think the best test would be to find an example where--
SPEAKER 5: [INAUDIBLE]
JUAN MALDACENA: What?
SPEAKER 5: [INAUDIBLE]
JUAN MALDACENA: Yeah, some example where we know both sides. So an analytic continuation from AdS is the bulk. So the bulk equation, Einstein's equations around Anti De-Sitter, or around De-Sitter, are very closely related. If you know the solutions to one, you know the solutions to the other.
But this analytic continuation doesn't work at the level of the boundary theory. We don't know what analytic continuation to do with the boundary theory to go from one to the other. Except in the case that Harmon and company studied. In that case, one can understand this analytic continuation at the boundary level.
SPEAKER 1: All right. So before we thank Juan again, let me remind you that the lectures will continue with a technical seminar tomorrow and a public lecture on Wednesday.
Thanks very much.
[APPLAUSE]

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Quantum mechanics is important for determining the geometry of spacetime. Juan Maldacena of the Institute for Advanced Study reviews the role of quantum fluctuations that determine the large scale structure of the universe, September 22, 2014, as part of the Department of Physics Bethe Lecture Series.